OWL FA: A Metamodeling Extension of OWL DL(∗) Jeff Z. Pan,1 Ian Horrocks1 and Guus Schreiber2 1 School of Computer Science, University of Manchester, UK {pan, horrocks}@cs.man.ac.uk 2 Computer Science Department, Free University Amsterdam, Netherlands schreiber@cs.vu.nl Abstract. Recent research has shown that the semantics of the standard SW annotation language RDF (as well as its ontological extension RDFS) and that of the standard SW ontology language OWL DL are not compatible with each other. Pan and Horrocks [5] propose a sub-language of RDFS, called RDFS(FA), which provides a clear connection between RDF and OWL DL. This pa- per proposes OWL FA, an extension of OWL DL with the metamodeling architecture of RDFS(FA). It also investigates some reasoning tasks for OWL FA. 1 Introduction The OWL Web Ontology Language is a W3C recommendation for expressing ontologies in the Semantic Web. There are three sub-language of OWL; namely, OWL Lite, OWL DL and OWL Full. In terms of both semantics and expressive power, OWL Lite and OWL DL are quite similar, while OWL DL and OWL Full are very different. OWL DL is regarded as the key OWL, which provides a good balance between expressive power and decidability. OWL Full is thought of as a ‘large’ OWL, which provides a free mixing of RDF and OWL, although it is clearly undecidable and has a non-standard semantics. A common user complaint about OWL DL is that it does not allow them to describe additional level(s) of classes and properties. It has been pointed out that if we just ignore the need for meta- classes and meta-properties, some users will simply not use OWL, and the whole effort could become a failure [9]. Example 1. A well known example [10] of using meta-classes and meta-properties is the WordNet ontology: WordNet synsets (the basic WordNet concepts) are represented as instances of the meta-class LexicalConcept. WordNet hyponym relations (the subclass relations in WordNet) are represented as tuples of the meta-property hyponymOf relation between instances of wns:LexicalConcept. These can be represented in the following RDFS statements, but not in OWL DL. wns:LexicalConcept rdfs:subClassOf rdfs:Class. wns:hyponymOf rdfs:subPropertyOf rdfs:subClassOf; rdfs:domain wns:LexicalConcept ; rdfs:range wns:LexicalConcept . wnc:100002086 wns:hyponymOf wnc:100001740 . Furthermore, we want to express that wns:hyponymOf is a transitive property: Trans(wns:hyponymOf), so that if hwnc:100002086, wnc:100001740i : wns:hyponymOf hwnc:100001740, wnc:100001923i : wns:hyponymOf, we can entail that wnc:100002086 is a wns:hyponymOf wnc:100001923. (∗) This work is partially supported by the FP6 Network of Excellence EU project Knowledge Web (IST-2004- 507842). OWL Full can be regarded as a not so successful attempt at integrating RDF with OWL DL. Firstly, there are at least three known problems in extending the RDF(S) Model Theory (RDF MT) [1] with OWL constructors [6, 7, 2]. Due to these problems, it is unknown whether the OWL Full semantics could give a coherent meaning to OWL Full ontologies; i.e., there may be OWL Full ontologies for which the semantics would not be well defined. Secondly, there is a serious mismatch between the semantics of OWL DL and OWL Full because OWL Full disagrees with OWL DL on valid OWL DL ontologies. More specifically, for two OWL DL ontologies O1 and O2 , O1 OWL Full-entails O2 does not imply that O1 OWL DL-entails O2 [8]. Furthermore, it has been shown that the metamodeling architecture of OWL Full also contributes to its undecidability [3]. In this paper, we propose OWL FA, an extension of OWL DL with the metamodeling architecture of RDFS(FA), which is a sub-language of RDFS that provides a clear connection with OWL DL. Intu- itively, RDFS(FA) stratifies the one-layer metamodeling architecture into a multiple-layer metamodeling architecture, so as to overcome the problem of dual roles that RDFS has. The satisfiability-preserving bi-directional mapping between RDFS(FA) axioms in 0-1 strata and OWL DL axioms suggests that we could extend OWL DL with the metamodeling architecture of RDFS(FA) [4]. 2 OWL FA Intuitively, OWL FA introduces a stratum number in class constructors and axioms to indicate the strata they belong to. Let i ≥ 0 be an integer. OWL FA consists of an alphabet of distinct class names VCi (for stratum i), datatype names VD , abstract property names VAPi (for stratum i), datatype property names VDP and individual (object) names (VI ); together with a set of constructors (with subscriptions) to construct OWL FA-classes and OWL FA-properties. OWL FA has a model theoretic semantics, which is defined in terms of interpretations. Given an OWL FA alphabet V, a set of built-in datatype names B ⊆ VD and an integer k ≥ 1, an OWL FA interpretation is a pair J = (∆J , ·J ), where ∆J is the domain (a non-empty set) and ·J is the interpretation function, which satisfy the following conditions (where 0 ≤ i ≤ k): 1. ∆J = J J S 0≤i≤k−1 ∆A i ∪ ∆D , where ∆A i is the domain for stratum i and ∆D is the datatype domain; J ∆A J ∆A J ×∆A J 2. ∆A i+1 = 2 i ∪2 i i and ∆D ∩ ∆A J = ∅; i 3. ∀a ∈ VI : a ∈ ∆A 0 and ∀C ∈ VCi+1 : CJ ⊆ ∆A J J J i ; J J J J J 4. ∀R S ∈ VAPi+1 : R ⊆ ∆A i × ∆A i and ∀T ∈ VDP : T ⊆ ∆A 0 × ∆D ; 5. ∀d∈B V (d) ⊆ ∆D , where V (d) is the value space of d; 6. ∀d ∈ VD , if d ∈ B, then3 (a) dJ = V (d), where V (d) is the value space of d, (b) if v ∈ L(d), then (“v”ˆˆd)J = L2V (d)(v), where L(d) is lexical space of d and L2V (d) is the lexical-to-value mapping of d, (c) if v 6∈ L(d), then (“v”ˆˆd)J is undefined; otherwise, dJ ⊆ ∆D and “v”ˆˆd ∈ ∆D . In the rest of the paper, we assume that i is an integer such that 1 ≤ i ≤ k. The interpretation function can be extended to give semantics to OWL FA-properties and OWL FA-classes. Let RN ∈ VAPi be an abstract property name in stratum i and R an abstract property in stratum i. Valid OWL FA abstract properties are defined by the abstract syntax: R ::= RN | R− , where for some x, y ∈ ∆A J i−1 , hx, yi ∈ RJ iff hy, xi ∈ R−J . Valid OWL FA datatype properties are datatype property names. Now we define the OWL FA-class descriptions. Let CN ∈ VCi be an atomic class name in stratum i, R an OWL FA-property in stratum i, o ∈ VI an individual, T ∈ VDP a datatype property name, and C, D OWL FA-classes in stratum i. Valid OWL FA-classes are defined by the abstract syntax: C ::= >i | ⊥ | CN | ¬i C | C ui D | C ti D | {o} | ∃i R.C | ∀i R.C | 6i nR | >i nR (if i = 1) ∃1 T.d | ∀1 T.d | 61 nT | >1 nT Constructor DL Syntax Semantics top >i ∆A J i−1 bottom ⊥ ∅ concept name CN CNJ ⊆ ∆A J i−1 general negation ¬i C ∆A J i−1 \ C J conjunction C ui D C J ∩ D J disjunction C ti D C J ∪ D J nominals {o} {o}J = {oJ } exists restriction ∃i R.C {x ∈ ∆A J J i−1 | ∃y.hx, yi ∈ R ∧ y ∈ C } J value restriction ∀i R.C {x ∈ ∆A J i−1 | ∀y.hx, yi ∈ R J → y ∈ C J } atleast restriction >i mR {x ∈ ∆A J i−1 | ]{y | hx, yi ∈ R J } ≥ m} atmost restriction 6i mR {x ∈ ∆A J J i−1 | ]{y | hx, yi ∈ R } ≤ m} datatype exists restriction ∃1 T.d {x ∈ ∆A 0 | ∃t.hx, ti ∈ T ∧ t ∈ dJ } J J datatype value restriction ∀1 T.d {x ∈ ∆A J0 | ∀t.hx, ti ∈ T J → t ∈ dJ } J J datatype atleast restriction >1 mT {x ∈ ∆A 0 | ]{t | hx, ti ∈ T } ≥ m} datatype atmost restriction 61 mT {x ∈ ∆A J J 0 | ]{t | hx, ti ∈ T } ≤ m} Table 1. OWL FA classes The semantics of OWL FA-classes are presented in Table 1 (page 3). C is satisfiable iff there exist an interpretation J s.t. C J 6= ∅; C subsumes D iff for every interpretation J we have C J ⊆ DJ . An OWL FA knowledge base Σ consists of Σ1 , . . . , Σk . Each Σi consists of a TBox Ti , an RBox Ri and an ABox Ai . Due to space limitation, we only provide details of OWL FA ABox here; it is obvious to extend traditional class and property axioms to meta-class axioms and meta-property axioms by introducing stratum numbers. Let a, b ∈ VI be individuals, C1 a class in stratum 1, R1 an abstract property in stratum 1, l a literal, T ∈ VD a datatype property, X, Y classes or abstract properties in stratum i, E a class in stratum i + 1 and S an abstract property in stratum i+1. An OWL FA ABox A1 is a finite set of individual axioms of the following forms: a :1 C1 , called class assertions, ha, bi :1 R1 , called abstract property assertions, ha, li :1 T , called datatype property assertions, a = b, called individual equality axioms and , a 6= b, called individual inequality axioms. An interpretation J satisfies a :1 C1 if aJ ∈ C1J ; it satisfies ha, bi :1 R1 if haJ , bJ i ∈ R1J ; it satisfies ha, li :1 T if haJ , lJ i ∈ T J ; it satisfies a = b if aJ = bJ ; it satisfies a 6= b if aJ 6= bJ . An OWL FA ABox Ai is a finite set of axioms of the following forms: X : E, called meta-class assertions, hX, Y i : R, called meta-property assertions, or X =i−1 Y , called meta individual equality axioms. An interpretation J satisfies X : E if X J ∈ E J ; it satisfies hX, Y i : R if hX J , Y J i ∈ RJ ; it satisfies X =i−1 Y if X J = Y J . Note that there are no meta-individual inequality axioms in OWL FA; it is more intuitive for users to apply disjoint class axioms. According to the above definition, it is obvious that Σ1 is a SHOIN (D) knowledge base, and Σ2 , . . . , Σk are SHIQ knowledge bases. Note that classes and property names in Σi are treated as individual names in Σi+1 ; therefore, class and property equality axioms in Σi can act as individual equality axioms in Σi+1 . On the other hand, individual equalities explicitly asserted and implicitly entailed by number restrictions in Σi+1 can act as class and property equality axioms in Σi . Definition 1. Let Σ = hΣ1 , . . . , Σk i be an OWL FA knowledge base, where each of Σ1 , . . . , Σk is consistent. Σ ∗ =hΣ1∗ , . . . , Σk∗ i, called the explicit knowledge base, is constructed by making all the implicit atomic class axioms, atomic property axioms and individual equality axioms explicit.  3 To simplify the presentation, we do not distinguish datatype names and datatype URIrefs here. As we have a finite set of vocabulary, we have the following Lemma. Lemma 1. Given an OWL FA knowledge base Σ =hΣ1 , . . . , Σk i. Σ ∗ can be constructed from Σ in finite steps. Proof (sketch): When k = 1, it is easy to show that we can calculate the explicit knowledge base Σ1∗ in finite steps because the sets of names of classes (in stratum 1), roles (in stratum 1) and individuals are finite. When k > 1, let us assume that we can calculate the explicit knowledge bases Σ10 , ..., Σi0 (where 1 ≤ i < k) from Σ1 , . . . , Σi in finite steps. We add all the class and property equality axioms in Σ 0i to Σi+1 . If the updated Σi+1 is consistent, we can make the implicit individual equality axioms (if any) explicit and add new class and property equality axioms into Σ 0i . According to our assumption, we can calculate Σ 001 , ..., Σ 00i in finite steps. As the individual names in Σi+1 are finite, we can calculate the explicit knowledge bases Σ1∗ , . . . , Σi+1 ∗ in finite steps. Theorem 1. Given an OWL FA knowledge base Σ =hΣ1 , . . . , Σk i and a class C in stratum i, C is satisfiable w.r.t. Σ iff C is satisfiable w.r.t. Σi∗ . Theorem 2. Given an OWL FA knowledge base Σ =hΣ1 , . . . , Σk i. Σ is satisfiable iff each Σi∗ (1 ≤ i ≤ k) is satisfiable. Theorem 2 indicates we can reduce the OWL FA-knowledge base satisfiability problem to the OWL DL-knowledge base satisfiability problem. 3 Related Work [3] also provides two alternative metamodeling approaches for OWL DL, i.e., the context approach and the HiLog approach. In the context approach, the names for classes, properties and individuals are not distinct and are interpreted depending on the context; i.e., they are interpreted by class interpretation functions, property interpretation functions and individual interpretation functions, respectively. The HiLog approach is closer to the spirit of OWL Full metamodeling. Datatypes are not covered in these two approaches. We now use an example in [3] to illustrate some of the differences among the above two approaches and our approach. Let us consider the following knowledge base4 Σ ={ Harry :1 Eagle, Harry :1 ¬Aquila, Eagle =1 Aquila}. In the context approach, since Eagle and Aquila as concepts and as individuals are independent, Σ is satisfiable. In the HiLog approach, it is not satisfiable because Eagle and Aquila are interpreted as the same object, let us call it a, and Harry cannot be both in and not in the concept extension of a. In OWL FA, Σ is unsatisfiable because the meta-individual equality axiom Eagle =1 Aquila indicates two concepts Eagle and Aquila are equivalent, and HarryJ cannot be both in and not in EagleJ . 4 Conclusion and Outlook In this paper, we propose the OWL FA ontology language as a metamodeling extension of OWL DL, using the metamodeling architecture of RDFS(FA), which is very similar to that of UML. The syntax of OWL FA is very similar to that of OWL DL; the former introduces a stratum number to attach to OWL FA class constructors and axioms. The semantics of OWL FA is a natural extension of that of OWL DL, dividing the abstract domain into k sub-domains for k strata. We have shown that OWL FA is decidable, and its basic reasoning tasks can be reduced to that of OWL DL. In the future, we plan to implement the construction of extended knowledge bases so that we can use OWL DL reasoners to reason with OWL FA ontologies, and to evaluate it with, for example, the WordNet ontology. 4 In [3], the subscripts are not used. Bibliography [1] Patrick Hayes. RDF Semantics. Technical report, W3C, Feb 2004. W3C recommendation, http://www.w3.org/TR/rdf-mt/. [2] Ian Horrocks, Peter F. Patel-Schneider, and Frank van Harmelen. From SHIQ and RDF to OWL: The making of a web ontology language. Journal of Web Semantics, 1(1):7–26, 2003. [3] Boris Motik. On the Properties of Metamodeling in OWL. 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